WORKED PROBLEMS

Problem 1

A student examines a thin section of an onion-root tip and records the number of cells that are in each stage of the cell cycle. She observes 94 cells in interphase, 14 cells in prophase, 3 cells in prometaphase, 3 cells in metaphase, 5 cells in anaphase, and 1 cell in telophase. If the complete cell cycle in an onion-root tip requires 22 hours, what is the average duration of each stage in the cycle? Assume that all cells are in the active cell cycle (not G₀).

Solution Strategy

What information is required in your answer to the problem?

The average duration of each stage of the cell cycle.

What information is provided to solve the problem?

• The numbers of cells in different stages of the cell cycle.
• A complete cell cycle requires 22 hours.

For help with this problem, review:

The Cell Cycle and Mitosis in Section 2.2.

Solution Steps

This problem is solved in two steps. First, we calculate the proportions of cells in each stage of the cell cycle, which correspond to the amount of time that an average cell spends in each stage. For example, if cells spend 90% of their time in interphase, then, at any given moment, 90% of the cells will be in interphase. The second step is to convert the proportions into lengths of time, which is done by multiplying the proportions by the total time of the cell cycle (22 hours).

STEP 1 Calculate the proportion of cells at each stage.

The proportion of cells at each stage is equal to the number of cells found in that stage divided by the total number of cells examined:

STEP 2 Determine the average duration of each stage.

To determine the average duration of each stage, multiply the proportion of cells in each stage by the time required for the entire cell cycle:

Problem 2

A cell in G₁ of interphase has 8 chromosomes. How many chromosomes and how many DNA molecules will be found per cell as this cell progresses through the following stages: G₂, metaphase of mitosis, anaphase of mitosis, after cytokinesis in mitosis, metaphase I of meiosis, metaphase II of meiosis, and after cytokinesis of meiosis II?

Solution Strategy

What information is required in your answer to the problem?

The number of chromosomes and number of DNA molecules present per cell at different stages of the cell cycle and meiosis.

What information is provided to solve the problem?

• A cell in G₁ has 8 chromosomes.
• Different stages of the cell cycle and meiosis.

For help with this problem, review:

Connecting Concepts: Counting Chromosomes and DNA Molecules in Section 2.2.

Solution steps

Remember the rules about counting chromosomes and DNA molecules: (1) to determine the number of chromosomes, count the functional centromeres; (2) to determine the number of DNA molecules, determine whether sister chromatids exist. If sister chromatids are present, the number of DNA molecules is 2 x the number of chromosomes. If the chromosomes are unreplicated (don’t contain sister chromatids), the number of DNA molecules equals the number of chromosomes. Think carefully about when and how the numbers of chromosomes and DNA molecules change in the course of mitosis and meiosis.

The number of DNA molecules increases only in the S phase, when DNA replicates; the number of DNA molecules decreases only when the cell divides. Chromosome number increases only when sister chromatids separate in anaphase of mitosis and in anaphase II of meiosis (homologous chromosomes, not chromatids, separate in anaphase I of meiosis). Like the number of DNA molecules, chromosome number is reduced only by cell division.

Let’s now apply these principles to the problem. A cell in G₁ has 8 chromosomes, and sister chromatids are not present; so 8 DNA molecules are present in G₁. DNA replicates in the S phase and now each chromosome consists of two chromatids; so, in G₂, 2 × 8 = 16 DNA molecules are present per cell. However, the two copies of each DNA molecule remain attached at the centromere; so there are still only 8 chromosomes present. As the cell passes through prophase and metaphase of the cell cycle, the number of chromosomes and the number of DNA molecules remain the same; so, at metaphase, there are 16 DNA molecules and 8 chromosomes. In anaphase, the chromatids separate and each becomes an independent chromosome; at this point, the number of chromosomes increases from 8 to 16. This increase is temporary, lasting only until the cell divides in telophase or subsequent to it. The number of DNA molecules remains at 16 in anaphase. The number of DNA molecules and chromosomes per cell is reduced by cytokinesis after telophase, because the 16 chromosomes and DNA molecules are now distributed between two cells. Therefore, after cytokinesis, each cell has 8 DNA molecules and 8 chromosomes, the same numbers that were present at the beginning of the cell cycle.

Now, let’s trace the numbers of DNA molecules and chromosomes through meiosis. At G₁, there are 8 chromosomes and 8 DNA molecules. The number of DNA molecules increases to 16 in the S phase, but the number of chromosomes remains at 8 (each chromosome has two chromatids). The cell therefore enters metaphase I having 16 DNA molecules and 8 chromosomes. In anaphase I of meiosis, homologous chromosomes separate, but the number of chromosomes remains at 8. After cytokinesis, the original 8 chromosomes are distributed between two cells; so the number of chromosomes per cell falls to 4 (each with two chromatids). The original 16 DNA molecules also are distributed between two cells; so the number of DNA molecules per cell is 8. There is no DNA synthesis in interkinesis, and each cell still maintains 4 chromosomes and 8 DNA molecules through metaphase II. In anaphase II, the two chromatids of each chromosome separate, temporarily raising the number of chromosomes per cell to 8, whereas the number of DNA molecules per cell remains at 8. After cytokinesis, the chromosomes and DNA molecules are again distributed between two cells, providing 4 chromosomes and 4 DNA molecules per cell. These results are summarized in the following table: